Manipulation of Temporal Localized Structures in a VECSEL With Optical Feedback

Abstract

We analyze both theoretically and experimentally the impact of optical feedback on the dynamics of external-cavity mode-locked semiconductor lasers (VECSELs) operated in the long cavity regime [1]. In particular, by choosing certain ratios between the cavity round-trip time and the feedback delay, we show that feedback acts as a solution discriminator that either reinforces or hinders the appearance of one of the multiple harmonic arrangements of temporal localized structures. For the theoretical modeling, the delayed differential equation model [2] is extended by a term describing the optical feedback. Denoting by A the amplitude of the optical field, G the gain, and Q the saturable losses, the model reads\begin{align*} & \frac{{\dot A}}{\gamma } = \sqrt \kappa \exp \left[ {\frac{{1 - i{\alpha _g}}}{2}G\left( {t - {\tau _c}} \right) - \frac{{1 - i{\alpha _a}}}{2}Q\left( {t - {\tau _c}} \right)} \right]A\left( {t - {\tau _c}} \right) - A(t) + \eta {e^{i{{\Omega }}}}A\left( {t - {\tau _f}} \right),\tag{1} \\ & \dot G = {g_0} - {{\Gamma }}G - {e^{ - Q}}\left( {{e^G} - 1} \right)|A{|^2},\quad \dot Q = {q_0} - Q - s\left( {1 - {e^{ - Q}}} \right)|A{|^2}\tag{2}\end{align*}where time has been normalized to the SA recovery time, α <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</inf> and α <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</inf> are the linewidth enhancement factors of the gain and absorber sections, respectively, κ the fraction of the power remaining in the cavity after each round- trip, g <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> the pumping rate, Γ the gain recovery rate, q <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> is the value of the unsaturated losses which determines the modulation depth of the SA, τ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</inf> the round-trip time in the cavity, s the ratio of the saturation energy of the SA and of the gain sections and γ is the bandwidth of the spectral filter, η is the feedback rate, Ω is the feedback phase and τ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</inf> the round-trip time in the feedback loop. For the experimental realization, the gain medium consisting in 6 quantum wells embedded between a bottom totally reflective Bragg mirror and a top partially reflective Bragg mirror (½VCSEL) was considered; the ½VCSEL was then placed in an external cavity that was closed by a fast semiconductor saturable absorber mirror (SESAM) to operate the laser in the passive mode-locked regime (see fig. 1). In the absence of the optical feedback such a laser can generate so-called harmonic mode-locked solutions denoted by HML <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</inf> where n is the number of equally spaced pulses per round-trip. The optical feedback induces that each pulse is followed by a train of small copies of itself. The size and position of this echo can be controlled via the feedback rate and the delay time, respectively. When an echo is placed close to the leading edge of another pulse, the pulse experiences less amplification by the gain medium as it is already depleted by the echo. This interaction can lead to the destruction of the main pulse. In consequence, the system settles on a solution where pulses and echos are well-separated and thus, do not interact. This mechanism is visualized in fig. 2 where a HML <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</inf> and HML <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> are exposed to optical feedback and therefor, the output changes to a HML <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</inf> and fundamental mode-locked solution, respectively. This behaviour was also observed in the experiment. In the further steps, a detailed bifurcation analysis is conducted where the influence of the position of the echo is investigated.